suplement o cap 5

7/25/2019 Suplement o Cap 5

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Supplement 5. Gaussian Integrals

An apocryphal story is told of a math major showing a psychology majorthe formula for the infamous bell-shaped curve or gaussian, which purports

to represent the distribution of intelligence and such:

The formula for a normalized gaussian looks like this:

(x) = 1

2ex

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The psychology student, unable to fathom the fact that this formula con-tained , the ratio between the circumference and diameter of a circle,asked Whatever does have to do with intelligence? The math student

is supposed to have replied, If your IQ were high enough, you would un-derstand! The following derivation shows where the comes from.

Laplace (1778) proved that

ex2

dx=

(1)

This remarkable result can be obtained as follows. Denoting the integralbyI, we can write

I2 =

ex2

dx2

=

ex2

dx

ey2

dy (2)

where the dummy variable y has been substituted for xin the last integral.The product of two integrals can be expressed as a double integral:

I2 =

e(x2+y2) dxdy

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The differentialdx dyrepresents an elementof area in cartesian coordinates,with the domain of integration extending over the entire xy-plane. Analternative representation of the last integral can be expressed in planepolar coordinates r, . The two coordinate systems are related by

x= r cos , y=r sin (3)

so that

r2 =x2 +y2 (4)

The element of area in polar coordinates is given by r dr d, so that thedouble integral becomes

I2 = 0

20

er2

r dr d (5)

Integration over gives a factor 2. The integral over r can be done afterthe substitutionu= r2, du= 2r dr:

0

er2

r dr= 12

0

eu du= 12 (6)

Therefore I2 = 2 12 and Laplaces result (1) is proven.A slightly more general result is

ex2

dx=

1/2(7)

obtained by scaling the variable x to

x.We require definite integrals of the type

xn ex2

dx, n= 1, 2, 3 . . . (8)

for computations involving harmonic oscillator wavefunctions. For odd n,the integrals (8) are all zero since the contributions from{, 0} exactlycancel those from{0,}. The following stratagem produces successive

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integrals for even n. Differentiate each side of (7) wrt the parameter andcancel minus signs to obtain

x2 ex2

dx= 1/2

2 3/2 (9)

Differentiation under an integral sign is valid provided that the integrandis a continuous function. Differentiating again, we obtain

x4 ex2

dx= 3 1/2

4 5/2 (10)

The general result is

xn ex2

dx= 135 |n1|1/2

2n/2(n+1)/2 , n= 0, 2, 4 . . . (11)

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suplement o cap 5

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